# A note on nontrivial intersection for selfmaps of complex Grassmann manifolds

## TL;DR

This paper proves that for certain selfmaps of complex Grassmann manifolds, there always exists a k-plane that intersects its image nontrivially, revealing a fundamental geometric property.

## Contribution

It establishes a new intersection property for selfmaps of complex Grassmann manifolds, extending understanding of their geometric structure.

## Key findings

- Existence of a k-plane with nontrivial intersection under any selfmap
- Applicable for 1<k<n in complex Grassmann manifolds
- Provides insight into the fixed point and intersection theory of these manifolds

## Abstract

Let $G(k,n)$ be the complex Grassmann manifold of $k$-planes in $\mathbb C^{k+n}$. In this note, we show that for $1<k<n$ and for any selfmap $f:G(k,n)\to G(k,n)$, there exists a $k$-plane $V^k\in G(k,n)$ such that $f(V^k)\cap V^k\ne \{0\}$.